Boundedness of l -Index and Completely Regular Growth of Entire Functions
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BOUNDEDNESS OF l-INDEX AND COMPLETELY REGULAR GROWTH OF ENTIRE FUNCTIONS A. I. Bandura1,2 and O. B. Skaskiv3
UDC 517.547.22
We study the relationship between the class of entire functions of completely regular growth of order ⇢ and the class of entire functions with bounded l-index, where l(z) = |z|⇢−1 + 1 for |z| ≥ 1. Possible applications of these functions in the analytic theory of differential equations are considered. We formulate three new problems on the existence of functions with given properties that belong to the differences of these classes. For the fourth problem, we obtain an affirmative answer, namely, we present sufficient conditions for an infinite product to be an entire function of completely regular growth of order ⇢ with unbounded l⇢ -index whose zeros do not satisfy the well-known Levin conditions (C) and (C 0 ). We also construct an entire function of completely regular growth of order ⇢ with unbounded l⇢ -index whose zeros do not satisfy the Levin conditions (C) and (C 0 ).
At present, the properties of entire functions of completely regular growth are studied fairly comprehensively (for the extensive bibliography dealing with this class of functions, see the monographs [1, 15, 22, 24, 30, 32]). However, there are numerous old problems in this field of complex analysis. As one of interesting problems, we can mention the Goldberg–Ostrovskii–Petrenko problem [14] (often abbreviated as the GOP-problem): Let w(n) + an−1 (z)w(n−1) + . . . + a1 (z)w0 + a0 (z)w = 0
(1)
be a given linear differential equation whose coefficients aj , 0 j n − 1, are entire function of completely regular growth. Is it true that each solution w(z) of Eq. (1) of finite order is a solution of completely regular growth? In [31], Petrenko formulated this problem without making the assumption that the coefficients aj (z) are entire functions of completely regular growth. He also gave the affirmative answer in the case where aj (z) are polynomials. In [14], Goldberg refuted this problems in Petrenko’s formulation. He showed that if f is an arbitrary entire function with zeros whose orders do not exceed n − 1, then it satisfies a linear differential equation of order n of the form (1), where aj , 0 j n − 1, are entire functions. Therefore, Goldberg and Ostrovskii corrected Petrenko’s formulation by assuming that the coefficients aj , 0 j n − 1, satisfy the conditions of completely regular growth. Note that, despite more than 30 years of investigations, this problem remains open [23]. We also note [33] that each entire solution of a linear differential equation of order n with constant coefficients is a function of bounded index. However, the entire functions of bounded l-index have properties that reveal their similarity to functions of completely regular growth. Let l : C ! R+ be a given positive continuous function, where R+ = (0, +1). An entire function f is called a function of bounded l-index [27] if there exists 1
Ivano-Frankivs’k National Oil and Gas Technical University, Ivano-Frankivs’k, Ukraine; e-m
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